Transcript No Slide Title

Interplay between disorder and interactions
in two dimensions
Sveta Anissimova
Sergey Kravchenko
(presenting author)
A. Punnoose
A. M. Finkelstein
Teun Klapwijk
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One-parameter scaling theory for non-interacting electrons:
the origin of the common wisdom “all states are localized in 2D”
d(lnG)/d(lnL) = b(G)
QM interference
1
b
0
-1
-2
G ~ Ld-2 exp(-L/Lloc)
MIT
metal (dG/dL>0) Ohm’s law in d dimensions
insulator
insulator
3D
2D
insulator
(dG/dL<0)
1D
-3
-4
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ln(G)
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Abrahams, Anderson, Licciardello,
and Ramakrishnan, PRL 42, 673
(1979)
However, the existence of the quantum Hall effect is
inconsistent with this prediction
Solution (Pruisken, Khmelnitskii…):
two-parameter (sxx, sxy) scaling theory
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Do the electron-electron interactions modify the
“all states are localized in 2D at B=0” paradigm?
(what happens to the Anderson transition
in the presence of interactions?)
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Corrections to conductivity due to electron-electron interactions
in the diffusive regime (Tt < 1)
 always insulating behavior
However, later this result was shown to be incorrect
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Zeitschrift fur Physik B (Condensed Matter) -- 1984 -- vol.56, no.3, pp. 189-96
Weak localization and Coulomb interaction in disordered systems
Finkel'stein, A.M.
L.D. Landau Inst. for Theoretical Phys., Acad. of Sci., Moscow, USSR

s


ln 1  F0
e
s  2  lnTt   1  3  1 
s
2 
F
0


2
 


behavior when interactions are weak (0.45  F0s  0)
s
(

1

F
 Altshuler-AronovMetallic
behavior
when
interactions
are
strong
0  0.45)
Lee’s result
Finkelstein’s & CastellaniEffective strength of interactions grows as the temperature
decreases
DiCastro-Lee-Ma’s
term
 Insulating
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Same mechanism persists to ballistic regime (Tt > 1),
but corrections become linear in temperature
3F0s
e 2 k BTt 
s T  
1
   1  F0s




This is reminiscent of earlier Stern-Das Sarma’s result
e 2 k BTt
s T 
C (ns ) where C(ns)
 
 
<0
(However, Das Sarma’s calculations are not applicable to strongly interacting regime because
at r s>1, the screening length becomes smaller than the separation between electrons.)
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What do experiments show?
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Strongly disordered Si MOSFET
(Pudalov et al.)
 Consistent with the one-parameter scaling theory
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Clean Si MOSFET, much lower electron densities
Kravchenko, Mason, Bowker,
Furneaux, Pudalov, and
D’Iorio, PRB 1995
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In very clean samples, the transition is practically universal:
Klapwijk’s sample:
Pudalov’s sample:
6
resistivity r (Ohm)
10
5
10
11
4
(Note: samples from
different sources,
measured in different labs)
3
10
0
0.5
1
temperature T (K)
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-2
0.86x10 cm
0.88
0.90
0.93
0.95
0.99
1.10
10
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1.5
2
… in contrast to strongly disordered samples:
clean sample:
disordered sample:
6
5
10
resistivity
r
(Ohm)
10
 Clearly, one-parameter scaling
theory does not work
here
0.86x10 cm
11
4
10
0.88
0.90
0.93
0.95
0.99
1.10
3
10
-2
0
0.5
1
1.5
2
temperature T (K)
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Again, two-parameter scaling theory comes to the rescue
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Two parameter scaling
(Finkelstein, 1983-1984;
Castellani, Di Castro, Lee, and Ma, 1984;
Punnoose and Finkelstein, 2002; 2005)
finite r increases g2
while g2 reduces r
to all orders in
the interplay of disorder and r and
interaction g2 changes the trend and
gives non-monotonic R(T)
g2
cooperon
singlet
“triplet”

 1 g 2

dr
 r 2  nv  1  (4nv2  1)  (
) ln(1  g 2 )  1 
d
 g2


dg 2
(1  g 2 ) 2
r
d
2
  ln(1/ Tt ) , Tt  1
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disorder takes over
disorder

QCP
interactions
Punnoose and Finkelstein, Science
310, 289 (2005)
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
metallic phase stabilized
by e-e interaction
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Experimental test of the Punnoose-Finkelstein theory
First, one needs to ensure that the system is in the diffusive regime (Tt < 1).
One can distinguish between diffusive and ballistic regimes by studying
magnetoconductance:
B
s B, T    
T 
2
- diffusive: low temperatures, higher disorder (Tt < 1).
2
B
- ballistic: low disorder, higher temperatures (Tt > 1).
s B, T  
T
The exact formula for magnetoconductance (Lee and Ramakrishnan, 1982):
 0.091e 2
s B, T   4

  h
2
2

 g B   B 

  g 2 g 2  1  
 k
 T 

 B
  

2
 g B B 

  1
 k T 
 B

Low-field magnetoconductance in the diffusive regime yields
strength of electron-electron interactions
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Experimental results (low-disordered Si MOSFETs;
“just metallic” regime; ns= 9.14x1010 cm-2):
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Temperature dependences of the
resistance (a) and strength of interactions (b)
This is the first time effective strength of interactions
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has been seen
to depend on T
Experimental disorder-interaction flow diagram of the 2D electron liquid
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Experimental vs. theoretical flow diagram
(qualitative comparison b/c the 2-loop theory was developed for multi-valley systems)
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Quantitative predictions of the two-parameter scaling theory
for 2-valley systems
(Punnoose and Finkelstein, Phys. Rev. Lett. 2002)
rmax
r(T)
Solutions of the RG-equations:
a series of non-monotonic curves r(T). After
rescaling, the solutions are described by a single
universal curve:
ρ(T) = ρmax R(η)
Tmax
For a 2-valley system (like Si MOSFET),
metallic r(T) sets in when g2 > 0.45
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g2(T)
η = ρmax ln(Tmax /T)
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g2 = 0.45
rmax ln(T/Tmax)
Resistance and interactions vs. T
Note that the metallic behavior sets in when g2 ~ 0.45,
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exactly as predicted
by the RG theory
Comparison between theory (lines) and experiment (symbols)
(no adjustable parameters used!)
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Si-MOSFET vs. GaAs/AlGaAs heterostructures
Si-MOSFET advantages
 Moderately high mobility: There exists a diffusive window T < 1/t < EF; 1/t = 2-3 K
 Short range scattering: Anderson transition in a disordered Fermi Liquid (universal)
 Two-valley system: Effects of electron-electron interactions are enhanced (“critical” g2=0.45
vs. 2.04 in a single-valley system)
GaAs/AlGaAs:
 Ultra high mobility: Diffusive regime is hard to reach; 1/t < 100-200 mK
 Long range scattering: Percolation type of the transition?
 Very low density: Non-degeneracy effects; possible Wigner crystallization,..
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Conclusions:
 It is demonstrated, for the first time, that as a result of the interplay
between the electron-electron interactions and disorder, not only the
resistance but also the interaction strength exhibits a fan-like spread as
the metal-insulator transition is crossed.
 Resistance-interaction flow diagram of the MIT clearly reveals a
quantum critical point, as predicted by renormalization-group theory of
Punnoose and Finkelstein.
 The metallic side of this diagram is accurately described by the
renormalization-group theory without any fitting parameters. In
particular, the metallic temperature dependence of the resistance sets in
once g2 > 0.45, which is in remarkable agreement with RG theory.
The interactions between electrons stabilize the metallic state in
2D and lead to the existence of a critical fixed point
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